| L(s) = 1 | − 3.36·3-s + 105.·5-s − 123.·7-s − 231.·9-s − 121·11-s − 493.·13-s − 356.·15-s + 685.·17-s − 2.23e3·19-s + 414.·21-s − 549.·23-s + 8.09e3·25-s + 1.59e3·27-s − 4.37e3·29-s + 100.·31-s + 406.·33-s − 1.30e4·35-s − 215.·37-s + 1.65e3·39-s − 1.86e3·41-s − 6.78e3·43-s − 2.45e4·45-s − 2.88e4·47-s − 1.60e3·49-s − 2.30e3·51-s − 3.81e4·53-s − 1.28e4·55-s + ⋯ |
| L(s) = 1 | − 0.215·3-s + 1.89·5-s − 0.950·7-s − 0.953·9-s − 0.301·11-s − 0.809·13-s − 0.408·15-s + 0.574·17-s − 1.41·19-s + 0.205·21-s − 0.216·23-s + 2.59·25-s + 0.421·27-s − 0.966·29-s + 0.0187·31-s + 0.0650·33-s − 1.80·35-s − 0.0258·37-s + 0.174·39-s − 0.173·41-s − 0.559·43-s − 1.80·45-s − 1.90·47-s − 0.0956·49-s − 0.124·51-s − 1.86·53-s − 0.571·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + 121T \) |
| good | 3 | \( 1 + 3.36T + 243T^{2} \) |
| 5 | \( 1 - 105.T + 3.12e3T^{2} \) |
| 7 | \( 1 + 123.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 493.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 685.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.23e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 549.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.37e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 100.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 215.T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.86e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 6.78e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.88e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.81e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.38e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.35e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.79e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.69e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.14e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.90e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.73e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 2.24e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.35e4T + 8.58e9T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.16144081955264777741929924393, −10.04687119662480753245900612230, −9.591436321133139682119921623279, −8.439069397462842926362608559965, −6.70837414689329244005288626174, −5.97912414019679257103634158984, −5.09810275575260876593372668156, −3.02441837575455649741797459087, −1.96063236567800423849373117486, 0,
1.96063236567800423849373117486, 3.02441837575455649741797459087, 5.09810275575260876593372668156, 5.97912414019679257103634158984, 6.70837414689329244005288626174, 8.439069397462842926362608559965, 9.591436321133139682119921623279, 10.04687119662480753245900612230, 11.16144081955264777741929924393
